Question

question 3 (2 points)
(03.04)
calculate the average rate of change for the function ( f(x)=-x^{4}+4x^{3}-2x^{2}+x + 1 ), from ( x = 0 ) to ( x = 1 ).
o a 0
o b 1
o c 2
o d 7

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 0\), \(b=1\).

Step2: Calculate \(f(0)\)

Substitute \(x = 0\) into \(f(x)=-x^{4}+4x^{3}-2x^{2}+x + 1\).
\(f(0)=-(0)^{4}+4(0)^{3}-2(0)^{2}+0 + 1=1\).

Step3: Calculate \(f(1)\)

Substitute \(x = 1\) into \(f(x)=-x^{4}+4x^{3}-2x^{2}+x + 1\).
\(f(1)=-(1)^{4}+4(1)^{3}-2(1)^{2}+1 + 1=-1 + 4-2 + 1+1=3\).

Step4: Calculate the average rate of change

Using the formula \(\frac{f(1)-f(0)}{1 - 0}\), substitute \(f(1)=3\) and \(f(0)=1\).
\(\frac{3-1}{1-0}=\frac{2}{1}=2\).

Answer:

C. 2